The distributive property states that a common factor can be factored out or it may also be stated as

**"a factor can be distributed over a sum either from the right or from the left"**.

For any three

real numbers a, b and c the distributive property says a(b+c) = ab + ac. By using the property of Symmetry, we can rewrite it as ab + ac = a(b+c)

This latter form of the Distributive Property, ab + ac = a(b+c), is often used to simplify polynomials.

**Example:** 4x

^{2} + 7x

^{2} is a polynomial which can be simplified by using the distributive property as

4x

^{2} + 7x

^{2} = (4+7)x

^{2} = 11x

^{2}The algebraic terms with same variable raised to the same power are like terms.We have combined the like terms. The unlike terms cannot be combined. The result which is simplified represents the same real number as the original expression when the variable or variables are replaced by real numbers.

The expressions that represent the same real numbers when variables are replaced by real numbers are called as Equivalent Expressions.

There a few forms of the Distributive Property that are necessary to simplify the polynomial expressions

1. -(a + b) = -1(a + b) = -a - b

2. -(a - b) = -1(a - b) = -a + b

3. (a + b)(c + d) = a(c + d) + b(c + d)

The like terms within any bracket must be combined as far as possible before the brackets are removed.

Again ,If a, b and c are any three integers, then,

**a (b + c) = (a × b) + (a × c) [right distributive law]**

**(b + c) a = (b x a) + (c x a) [left distributive law]**In the left hand side of the above equation, ‘a' multiplies with the sum of ‘b' and ‘c' on the right hand side, it multiplies ‘b' and the ‘c' individually, with the results added afterwards. Because both expression give the same value, we can say that multiplication by ‘a' distributes over the addition of b and c. Since if we put any integer in place of ‘a', ‘b', and ‘c' above, and still have the same value of the expression, we say that the multiplication of integers distributes over the addition of integers.

Consider the following:

1) 5 (2 + 3) = (5 × 2) + (5 × 3)

Consider the left hand side (L.H.S), 5 (2 + 3) [Here the left hand side of the equation, 5 multiplies the sum of 2 and 3]

= 5(5)

= 25

Now consider right hand side (R.H.S), (5 × 2) + (5 × 3) [here we observe that in the right hand side, 5 multiplies 2 and 3 individually, with the results added afterwards].

= (10) + (15) = 25

Therefore, L.H.S = R.H.S

2) 4 (1 + 3) = (4 × 1) + (4 × 3)

Consider the left hand side (L.H.S), 4 (1 + 3) [Here in the left hand side of the equation, 4 multiplies the sum of 1 and 3]

= 4(4)

= 16

Now consider right hand side (R.H.S), (4 × 1) + (4 × 3) [here we observe that in the right hand side, 4 multiplies 1 and 3 individually, with the results added afterwards].

= (4) + (12) = 16

Therefore, L.H.S = R.H.S